Cahill's Extra Quantum Gravity Term
My comments are in italics
Reginald Cahill has come up with a new derivation for the way gravity acts [1,2], based on looking at a generalized vector velocity field, which gives an extra term proportional to the fine structure constant alpha. This connects his theory to quantum effects through Planck's constant, and gives some rationale for the experimental fact that gravity acts almost instantaneously rather than propagating at light speed. But the biggest effect of the new term is to create conditions under which gravitational attraction can vary somewhere between 1-over-r-squared and 1-over-r under certain conditions. This provides a new explanation for: 1) the apparent decrease in gravitational acceleration in deep (a few km) bore holes in arctic ice; 2) an explanation for the apparent mass of the central black hole in various types of galaxies, and 3) an explanation for the rotation curves of spiral galaxies without resorting to theoretical dark matter. Cahill refers to all of these as black hole effects, while I will refer to them as extra mass effects.
Cahill begins: Our understanding of gravity is based on Newton's modeling of Kepler's phenomenological laws for the motion of the planets within the solar system. In this model Newton took the gravitational acceleration field to be the fundamental dynamical degree of freedom, which is determined by the matter distribution; essentially via the "universal inverse square law", i.e. Newton began with inverse square, so deviations to this would not be apparent.
If, rather than using an acceleration field, a vector velocity field is assumed to be fundamental to gravity, then we immediately find that "extra mass" effects arise as a space self-interaction dynamical effect, and that the observed correlation is simply that Mextra/M = alpha/2 for spherical systems, where alpha is the fine structure constant, 1/137.036, which contains Planck's constant giving gravity a quantum component. The extra mass is only 0.365% of the actual mass, so it would not ordinarily be observable except that it produces quite different dynamical behavior in some situations!
Let us investigate in a phenomenological way the consequences of using a vector velocity field v(r, t) to be the fundamental dynamical degree of freedom to model gravity. This form is mandated by Galilean covariance under change of observer. In terms of the velocity field, Newtonian gravity dynamics involves using r-dot to construct a rank-0 tensor that can be related to the matter density rho. The coefficient turns out to be the Newtonian gravitational constant G. This is clearly equivalent to the differential form of Newtonian gravity outside of a spherical mass M which gives the usual inverse square law
The simplest non-Newtonian dynamics involves the two rank-0 tensors constructed at 2nd order from partial(vi)/partial(xj). Hence the modeling of gravity now involves two gravitational constants G and alpha, with alpha being the strength of the self-interaction dynamics, but which was not apparent in the solar system dynamics. All the various experimental phenomena discussed herein imply that alpha is the fine structure constant 1/137 up to experimental errors.
When the matter density is confined to a sphere of radius R we find that the "extra mass" density is confined to that sphere, and that consequently g (r) has an inverse square law behavior outside of the sphere. We find inside radius R that the total "extra mass", to Order (alpha), is Mextra = M (alpha/2), independently of the matter density profile. This turns out to be a very useful property as knowledge of the density profile is then not required in order to analyze observational data.
Galaxies in General
The data for a number of galaxies is given in the paper, with the assumption that the extra mass is all in the central black hole.
These data, when plotted on a log-log scale, cluster around alpha/2, with deviations that depend on galaxy type that shows that some of the assumptions in the theory are only approximate in these cases.
Mextra/M, in particular, for globular clusters M15 and G1 and highly spherical "elliptical" galaxies M32, M87 and NGC 4374, show that this ratio lies close to the "alpha/2-line", where alpha is the fine structure constant 1/137. However for the presumably dynamically varying spiral galaxies their Mextra/M values do not cluster close to the alpha/2-line. Hence it is suggested that these spherical systems manifest minimal "black hole" dynamics. However these dynamics are universal, so that any spherical system must induce such a minimal black hole mode, but for which, outside of such a system, only the Newtonian inverse square law would be apparent.
This mode must also apply to the Earth, which is certainly a surprising prediction. However just such an effect has been manifested in measurements of g in mine shafts and bore holes since the 1980's. Data from these geophysical measurements give us a very accurate determination of the value of alpha. To understand this bore-hole anomaly we need to compute the expression for g (r) just beneath and just above the surface of the Earth. To lowest order in alpha, this gives Newton's "inverse square law" for r > R, but in which we see that the effective Newtonian gravitational constant is GN = (1+ alpha/2 )G, which is slightly different from the fundamental gravitational constant G. This is caused by the additional "extra mass". Inside the Earth a few kilometers we see a g (r) different from Newtonian gravity. The effect is that g decreases more slowly with depth than predicted by Newtonian gravity. Delta g (r) is found to be, to 1st order, linear in R - r. Delta g (r) = -2 pi alpha GN rho(R)(R - r) for r < R, and Delta g (r) = 0 for r > R.
Spiral Galaxy Rotation
We now consider the somewhat sparse density situation in which matter in-falls around an existing primordial black hole. There are measurements of the masses of various galaxies and their central black holes. Immediately we see some of the consequences of this time evolution: (1) the acceleration field falls off much slower than the Newtonian inverse square law, as this local in-fall would happen very rapidly; (2) the resultant in-flow would result in the matter rotating much more rapidly than would be predicted by Newtonian gravity; and (3) this would result in a spiral galaxy exhibiting non-Keplerian rotation of stars and gas clouds, via the "extra mass" effect.
We can determine the star orbital speeds for highly non-spherical galaxies in the asymptotic region by solving asymptotically where rho is approximately 0. The velocity field will be approximately spherically symmetric and radial; nearer in we would match such a solution to numerically determined solutions. We then compute circular orbital speeds giving the predicted "universal rotation-speed curve". Because of the alpha dependent part, this rotation-speed curve falls off extremely slowly with r, as is indeed observed for spiral galaxies. Thus, the essentially flat galaxy rotation is explained by an extra gravitation effect rather than by hypothetical dark matter. This also corresponds to Van Flandern's  assertion that such rotation could be explained if gravity varied as 1-over-r over the volume of the galaxy. This is illustrated in Fig. 3 for the spiral galaxy NGC 3198.
If gravity is not always 1-over-r-squared, and has a quantum component, then several previously unexplained phenomena can be explained without resorting to dark matter. This becomes another good argument against the validity of the Big Bang.
1) R. T. Cahill, "Black Holes in Elliptical and Spiral Galaxies and in Globular Clusters", Progress in Physics, 3, pp 51-56, October 2005.
2) R. T. Cahill, "Black Holes and Quantum Theory: The Fine Structure Constant Connection", Progress in Physics, 4, pp 44-50, October 2006.
3) T. Van Flandern, Dark Matter, Missing Planets, & New Comets, North Atlantic Books, Berkeley, California, 1998.